Video: Multiplying Positive Rational Expressions with Mixed Positive and Negative Integer Exponents

Which of the following is equal to (2/3)⁻³ × (2/3)⁵? [A] (2/3) ⁻¹⁵ [B] −(2/3)⁸ [C] 4/9 [D] 9/4 [E] 16/81

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Video Transcript

Which of the following is equal to two-thirds to the power of negative three multiplied by two-thirds to the power of five. Is it (A) two-thirds to the power of negative 15, (B) negative two-thirds to the power of eight, (C) four-ninths, (D) nine-quarters, or (E) 16 over 81?

In order to answer this question, we need to recall one of our laws of exponents or indices. 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏. When multiplying two terms with the same base, we can add the exponents. This means that our expression simplifies to two-thirds to the power of negative three plus five. Negative three plus five is equal to two. So, we are left with two-thirds squared.

When squaring a number, we multiply it by itself. And when multiplying two fractions, we multiply the numerators and denominators separately. Two multiplied by two is four, and three multiplied by three is nine. This is the same as squaring the numerator and then squaring the denominator. The correct answer is option (C). Two-thirds to the power of negative three multiplied by two-thirds to the power of five is equal to four-ninths.

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